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\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{复变函数第一章部分习题}
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\date{2024 年 3 月 11 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item [1.]   %Problem 01
设 $z=\frac{1-\sqrt{3}i}{2}$, 求 $|z|$ 及 $\mathrm{Arg}(z)$. 

\vspace{0.2cm}

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\item [3.]  %Problem 02
求方程 $z^3+z^2+z+1=0$ 的根，并将 $z^3+z^2+z+1$ 因式分解。

\vspace{0.2cm}

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\item [5.]  %Problem 03
设三个复数 $z_1,z_2,z_3$ 符合条件 $z_1+z_2+z_3=0$ 以及 $|z_1|=|z_2|=|z_3|=1$. 
证明这三点是一个内接于单位圆周的正三角形的顶点。

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\item [7.]  %Problem 04
设 $n>1$ 为正整数，证明 $$\prod_{k=1}^{n-1} \left(x^2-2x\cos\frac{k\pi}{n}+1 \right) = 
\frac{x^{2n}-1}{x^2-1}. $$

\vspace{0.2cm}

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\item [10.]  %Problem 05
求下述参数方程给出的曲线：
\begin{enumerate}
\item  $z=(1+i)t$. 
\item  $z=a\cos(t)+ib\sin(t)$. 
\item  $z=t+\frac{i}{t}$. 
\item  $z=t^2+\frac{i}{t^2}$. 
\end{enumerate}

\vspace{0.2cm}

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\item[11.]  %Problem 06
函数 $w=\frac{1}{z}$ 将 $z$ 平面上的下述曲线变成 $w$ 平面上的什么曲线？
\begin{enumerate}
\item  $x^2+y^2=4$. 
\item  $y=x$. 
\item  $x=1$. 
\item  $(x-1)^+y^2=1$. 
\end{enumerate} 

\vspace{0.2cm}

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\item[14.]  %Problem 07
设 $f(z)=e^x(\cos y + i\sin y)$, 其中 $z=x+iy$, 问当 $z\to\infty$ 时，$f(z)$ 有无极限，包括广义极限？

\vspace{0.2cm}

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\item[15.]  %Problem 08
函数 $f(z)=\frac{1}{1-z}$ 在单位圆 $|z|<1$ 内是否连续？是否一致连续？

\vspace{0.2cm}

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\item[H6.]  %Problem 09
设 $|z|=1$, 证明 $$\left\vert \frac{az+b}{\bar{b}z+\bar{a}} \right\vert =1. $$

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\item[H7.]  %Problem 10
已知正方形的两个相对的顶点坐标为 $z_1(0,-1)$ 和 $z_3(2,5)$, 求另两个顶点的坐标。
\vspace{0.2cm}


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\end{enumerate}


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\end{document}

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